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Magnetic Fields
and Forces
Magnetic Fields
• Every magnet, regardless of its shape, has two
“poles”, called “north” and “south”.
• These poles exert forces on each other in a manner
analogous to electric charges.
• The poles received their names from the behavior
of a magnet in the earth’s magnetic field.
• Difference: Electric charges can be “isolated”
while magnetic poles are always found in pairs.
The Discovery
of the Magnetic Field
In April of 1820, the Danish physicist Hans
Christian Oersted was giving a evening lecture in
which he was demonstrating the heating of a wire
when an electrical current passed through it. He
noticed that a compass that was on the table deflected
each time he made the current flow.
Until that time, physicists had considered electricity
and magnetism to be unrelated phenomena. Oersted
discovered that the “missing link” between electricity
and magnetism was the electric current.
Hans Christian Oersted
(1777- 1851)
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Serway & Jewett, Principles of Physics, 3rd ed.
Figure 22.1
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Current Effect on Compass
Place some compasses around a wire. When no current is flowing in the
wire, all compasses point north.
When current flows in the wire, the compasses point in a ring around the
wire.
The Right-Hand Rule: Grip the wire so that the thumb of your right hand
points in the direction of the current. Then your fingers will point in the
direction that the compasses point. This is the direction of the magnetic field
created by the current flow.
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Vector Conventions
Example: field around a current.
For discussions of magnetism, we will need a three-dimensional
perspective, but we will use two-dimensional diagrams when we can. To get
the 3rd dimension into a two-dimensional diagram, we will indicate vectors
into and out of a diagram by using crosses and dots, respectively.
Rule: a dot (•) means you are looking at the point of an arrow coming
toward you; a cross (×
×) means you are looking at the tail feathers of an arrow
going away from you.
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The Magnetic Field
Definition of the magnetic field:
(1) The magnetic field at each point is a
vector, with both a magnitude, which we
call the magnetic field strength B, and a
direction.
(2) A magnetic field is created at all points in
the space surrounding a current carrying
wire.
(3) The magnetic field exerts a force on
magnetic poles. The force on a north pole
is parallel to B, and the force on a south
pole is antiparallel to B.
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Magnetic Field Lines
The magnetic field can be graphically
represented as magnetic field lines, with the
tangent to a given field line at any point
indicating the local field direction and the
spacing of field lines indicating the local
field strength.
B-field lines
never cross.
B-field line spacing
indicates field
strong
strength
weak
B-field lines form
closed loops.
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Two Kinds of Magnetism?
We jumped from a discussion of the
magnetic effects of permanent bar magnets to
the magnetic fields produced by current
carrying wires.
Are there two distinct kinds of magnetism?
No. As we will see in the lectures that
follow, the two manifestations of magnetism
are actually two aspects of the same
fundamental magnetic force.
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Magnetic Fields
• Recall: the gravitational field g at some
point in space is the gravitational force
acting on a “test mass” divided by the test
mass, i.e.,
g = Fg/mo
• The electric field E at some point in space is
the electric force acting on a “test charge”
divided by the test charge, i.e.,
E = FE/qo
Magnetic Fields
• The magnetic field vector B (or “magnetic
induction” or “magnetic flux density”) is
now defined at some point in space in terms
of the magnetic force acting on an
appropriate “test object”.
• “Test Object” = a charged particle moving
with velocity v.
Magnetic Fields
q
θ
B
v
• The magnitude of the magnetic force, FB , is
proportional to the charge q, the speed v = | v | of
the particle, and the magnetic field B.
• | FB | also depends on θ.
Magnetic Fields
• When the charged particle moves parallel to the
magnetic field B, then FB = 0.
• When the velocity vector v makes an angle θ
with the magnetic field B, the magnetic force
acts in a direction perpendicular to both v and
B.
• The magnetic force on a negative charge is in
the direction opposite to the force on a positive
charge moving in the same direction.
Magnetic Force on a Charge
• If the velocity vector makes an angle θ with the
magnetic field, the magnitude of the magnetic
force is proportional to sin θ.
• All of these observations can be summarized by
using a special vector notation to write the
magnetic force:
FB = q v × B
• The product denoted by × is called the cross
product.
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Magnetic Force
A current consists of moving charges. Ampere’s experiment
implies that a magnetic field exerts a force on a moving charge. This
is true, although the exact form of the force relation was not
discovered until later in the 19th century. The force depends on the
relative directions of the magnetic field and the velocity of the
moving charge, and is perpendicular to both.
(
F = q v×B
)
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Magnetic Force on Moving Charges
Properties of the magnetic force:
1. Only moving charges experience the
magnetic force. There is no
magnetic force on a charge at rest
(v = 0) in a magnetic field.
2. There is no magnetic force on a
charge moving parallel (θ
θ = 0º) or
anti-parallel (θ = 180º) to a magnetic
field.
3. When there is a magnetic force, it is
perpendicular to both v and B.
4. The force on a negative charge is in
the direction opposite to v x B.
5. For a charge moving perpendicular
to B (θ
θ = 90º) , the magnitude of the
force is F=|q|vB.
(
F = q v×B
)
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Magnetic Fields
• The direction of the magnetic force FB
depends on the sign of the particle’s charge,
the direction of its velocity, and on the
direction of the magnetic field.
• The direction of the magnetic force FB is
given by the Right Hand Rule.
Serway & Jewett, Principles of Physics, 3rd ed.
Figure 22.4
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Serway & Jewett, Principles of Physics, 3rd ed.
Figure 22.3
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Example: Magnetic Force on an
Electron
A long straight wire carries a 10 A
current from left to right. An electron 10
cm above the wire is traveling to the
right with a speed of 1x107 m/s. The
magnetic field 10 cm away from the wire
is 2 x 10-4 T (we will learn to calculate
this later)
What is the magnitude and direction
of the force on the electron.
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Example: Finding Angle Between v and B
A proton moving at 4.0 x106 m/s through a magnetic field of
1.7 T experiences a magnetic force of magnitude 8.2x10-13 N.
What is the angle between the proton’s velocity and the
magnetic field?
Magnetic Fields
Differences between electric and magnetic forces on
charged particles.
• The electric force on a charged particle is independent
of the particle’s speed.
• The magnetic force only acts on a charged particle
when the particle is in motion.
• The electric force is always along or opposite to the
electric field.
• The magnetic force is perpendicular to the magnetic
field.
(FE = q E vs. FB= q v × B)
Magnetic Fields
Differences between electric and magnetic forces
on charged particles.
• The electric force does work in displacing a charged
particle. The magnetic force does no work when a
charged particle is displaced. A magnetic field can
change the direction but not the speed of a moving
charged particle.
Magnetic Fields
Units of B
The SI unit of B is
weber/sq. meter = Wb/m2 = tesla = T
= N/(C•m/s ) = 104 G (gauss)
Earth’s magnetic field
~ 0.5G = 0.5 x 10-4 T
Magnetic Forces on
Current-Carrying Wires
When a wire carries a current that is parallel or
anti-parallel to a magnetic field, there is no force
(because the charges move along field lines).
When a wire carries a current that is
perpendicular to a magnetic field, there is a force
on the wire perpendicular to the current and field.
q
q
qv
I=
=
=
∆t L / v L
F = qvB = ILB
Fwire = ILB
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Example: Magnetic Levitation
A 0.10 T uniform magnetic field is
horizontal, parallel to the floor. A 0.5 m long
segment of copper wire with a mass of 0.014
kg is also parallel to the floor and
perpendicular to the field. What current
through the wire in what direction will allow
the wire to “float” in the magnetic field?
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Serway & Jewett, Principles of Physics, 3rd ed.
Figure 22.15
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The Force between
Two Parallel Wires
Parallel wires carrying current in the same direction attract each other.
Parallel wires carrying current in opposite directions repel each other.
Bwire
µ 2I
= 0
4π d
We will derive this equation later, but for now it is
useful to be aware of it. Finding the force between
two wires is a common problem to solve.
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Ampere’s Experiment
When Ampere heard of Oersted’s results, he
reasoned that if a current produced a magnetic
effect, it might respond to a magnetic effect.
Therefore, he measured the force between two
parallel current-carrying wires.
He found that parallel currents create an
attraction between the wires, while anti-parallel
currents create repulsion.
André Marie Ampère
(1775 – 1836)
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Forces
on Current Loops
Parallel currents in loops attract.
Opposite currents in loops repel.
Magnetic poles attract or repel because
the moving charges in one current
producing the pole exert an attractive or
repulsive magnetic force on the moving
charges in the current producing the other
pole.
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Torques on Current Loops
Consider the forces on a current loop
carrying current I that is a square of length L
on a side that is in a uniform magnetic field B.
Its area vector makes an angle θ with B.
Ftop = − Fbottom
Ffront = − Fback
∑F = 0
τ = r × F = rF sin θ
τ = ( F )(d ⊥ )
To find the total torque (net torque), find the torque
from each segment and add all the torques together
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τ net = τ left segment + τ right segment
τ net
 w
 w
= ( IhB)   + ( IhB)  
2
2
τ net = ( IBhw)
τ net = ( IAB)
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If it is not in the position where the torque is
maximum, the perpendicular distance
changes to:
 w
d ⊥ =   sin θ
2
where θ is the angle between the normal to the
loop and the magnetic field.
so the net torque is given by:
τ net = τ left segment + τ right segment
τ net
 w
 w
= ( IhB)   sin θ + ( IhB)   sin θ
2
2
τ net = ( IBhw) sin θ
τ net = ( IAB) sin θ
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If there are multiple loops of wire, then each one
contributes the same torque to the loop, so you would
multiple the torque of a single loop by the number of loops
τ net = N ( IAB) sin θ
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An Electric Motor
We can use the torque of a loop in a magnetic field to make an electric motor. The
current through the loop passes through a commutator switch, which reverses the
current as the loop approaches the equilibrium position.
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Atomic Magnets
A plausible explanation for the magnetic
properties of materials is the orbital
motion of the atomic electrons. The
figure shows a classical model of an atom
in which a negative electron orbits a
positive nucleus. The electron's motion
is that of a current loop. Consequently,
an orbiting electron acts as a tiny
magnetic dipole, with a north pole and
a south pole.
However, the atoms of most elements contain many electrons. Unlike the
solar system, where all of the planets orbit in the same direction, electron orbits
are arranged to oppose each other: one electron moves counterclockwise for
each electron that moves clockwise. Thus the magnetic moments of individual
orbits tend to cancel each other and the net magnetic moment is either zero or
very small.
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